Results: Plots and Tables

This report contains different plots and tables that may be relevant for analysing the results. Observe:

• The hypercube \(Y_N\) is placed in, is approx. \([0, 10000m]^p\), since for each subproblem \(Y_N^s \subset [0, 10000]^p\). That is, more subproblems create a larger hypercube for the master problem and the volume (number of integer points) grows exponentially in \(p\).
• Given \(|Y_N|\), we can find the hypercube as \[[l_1, u_1] \times \cdots \times [l_p, u_p],\] where \(l_i = \min_{y \in Y_N}(y_i)\) and \(u_i = \max_{y \in Y_N}(y_i)\).
• The number of integer points in the hypercube is \[\prod_{i = 1}^p (u_i - l_i + 1)\]

Statistics for alg1

Given a problem consisting of \(m\) subproblems with \(Y_N^s\) given for each subproblem \(s\), we use a filtering algorithm to find \(Y_N\) (alg1).

The following instance/problem groups are generated given:

• \(p=2,\ldots, 5\). [4 options]
• \(m=2,\ldots 5\) where \(m\) is the number of subproblems. [4 options]
• All subproblems have the same method config or half have method u and l. [4 options]
• Five instances for each config. [5 options]

Status

1235/1280 problems have been solved, i.e. 45 remaining:

##  [1] "alg1-prob-5-100|100|100|100|100-uuull-5_1.json"
##  [2] "alg1-prob-5-100|100|100|100|100-uuull-5_2.json"
##  [3] "alg1-prob-5-100|100|100|100|100-uuull-5_3.json"
##  [4] "alg1-prob-5-100|100|100|100|100-uuull-5_4.json"
##  [5] "alg1-prob-5-100|100|100|100|100-uuull-5_5.json"
##  [6] "alg1-prob-5-200|200|200|200|200-mmmmm-5_1.json"
##  [7] "alg1-prob-5-200|200|200|200|200-mmmmm-5_2.json"
##  [8] "alg1-prob-5-200|200|200|200|200-mmmmm-5_3.json"
##  [9] "alg1-prob-5-200|200|200|200|200-mmmmm-5_4.json"
## [10] "alg1-prob-5-200|200|200|200|200-mmmmm-5_5.json"
## [11] "alg1-prob-5-200|200|200|200|200-uuull-5_1.json"
## [12] "alg1-prob-5-200|200|200|200|200-uuull-5_2.json"
## [13] "alg1-prob-5-200|200|200|200|200-uuull-5_3.json"
## [14] "alg1-prob-5-200|200|200|200|200-uuull-5_4.json"
## [15] "alg1-prob-5-200|200|200|200|200-uuull-5_5.json"
## [16] "alg1-prob-5-200|200|200|200|200-uuuuu-5_1.json"
## [17] "alg1-prob-5-200|200|200|200|200-uuuuu-5_2.json"
## [18] "alg1-prob-5-200|200|200|200|200-uuuuu-5_3.json"
## [19] "alg1-prob-5-200|200|200|200|200-uuuuu-5_4.json"
## [20] "alg1-prob-5-200|200|200|200|200-uuuuu-5_5.json"
## [21] "alg1-prob-5-300|300|300|300|300-lllll-5_1.json"
## [22] "alg1-prob-5-300|300|300|300|300-lllll-5_2.json"
## [23] "alg1-prob-5-300|300|300|300|300-lllll-5_3.json"
## [24] "alg1-prob-5-300|300|300|300|300-lllll-5_4.json"
## [25] "alg1-prob-5-300|300|300|300|300-lllll-5_5.json"
## [26] "alg1-prob-5-300|300|300|300|300-mmmmm-5_1.json"
## [27] "alg1-prob-5-300|300|300|300|300-mmmmm-5_2.json"
## [28] "alg1-prob-5-300|300|300|300|300-mmmmm-5_3.json"
## [29] "alg1-prob-5-300|300|300|300|300-mmmmm-5_4.json"
## [30] "alg1-prob-5-300|300|300|300|300-mmmmm-5_5.json"
## [31] "alg1-prob-5-300|300|300|300|300-uuull-5_1.json"
## [32] "alg1-prob-5-300|300|300|300|300-uuull-5_2.json"
## [33] "alg1-prob-5-300|300|300|300|300-uuull-5_3.json"
## [34] "alg1-prob-5-300|300|300|300|300-uuull-5_4.json"
## [35] "alg1-prob-5-300|300|300|300|300-uuull-5_5.json"
## [36] "alg1-prob-5-300|300|300|300|300-uuuuu-5_1.json"
## [37] "alg1-prob-5-300|300|300|300|300-uuuuu-5_2.json"
## [38] "alg1-prob-5-300|300|300|300|300-uuuuu-5_3.json"
## [39] "alg1-prob-5-300|300|300|300|300-uuuuu-5_4.json"
## [40] "alg1-prob-5-300|300|300|300|300-uuuuu-5_5.json"
## [41] "alg1-prob-5-50|50|50|50|50-uuull-5_1.json"     
## [42] "alg1-prob-5-50|50|50|50|50-uuull-5_2.json"     
## [43] "alg1-prob-5-50|50|50|50|50-uuull-5_3.json"     
## [44] "alg1-prob-5-50|50|50|50|50-uuull-5_4.json"     
## [45] "alg1-prob-5-50|50|50|50|50-uuull-5_5.json"

1235/1235 problems have 5 instances solved for each configuration. Configurations with lees that 5 solved:

## # A tibble: 9 × 5
## # Groups:   p, m, method [4]
##       p     m method spAveCard solved
##   <dbl> <dbl> <chr>      <dbl>  <int>
## 1     5     5 l            300      0
## 2     5     5 m            200      0
## 3     5     5 m            300      0
## 4     5     5 u            200      0
## 5     5     5 u            300      0
## 6     5     5 ul            50      0
## 7     5     5 ul           100      0
## 8     5     5 ul           200      0
## 9     5     5 ul           300      0

73/1235 have not been classified at all:

##  [1] "alg1-prob-4-200|200|200|200|200-mmmmm-5_1.json"
##  [2] "alg1-prob-4-200|200|200|200|200-mmmmm-5_2.json"
##  [3] "alg1-prob-4-200|200|200|200|200-mmmmm-5_3.json"
##  [4] "alg1-prob-4-300|300|300|300|300-lllll-5_4.json"
##  [5] "alg1-prob-4-300|300|300|300|300-mmmmm-5_1.json"
##  [6] "alg1-prob-4-300|300|300|300|300-mmmmm-5_2.json"
##  [7] "alg1-prob-4-300|300|300|300|300-mmmmm-5_3.json"
##  [8] "alg1-prob-4-300|300|300|300|300-mmmmm-5_4.json"
##  [9] "alg1-prob-4-300|300|300|300|300-mmmmm-5_5.json"
## [10] "alg1-prob-4-300|300|300|300|300-uuull-5_2.json"
## [11] "alg1-prob-4-300|300|300|300|300-uuull-5_3.json"
## [12] "alg1-prob-4-300|300|300|300|300-uuull-5_4.json"
## [13] "alg1-prob-4-300|300|300|300|300-uuull-5_5.json"
## [14] "alg1-prob-4-300|300|300|300|300-uuuuu-5_2.json"
## [15] "alg1-prob-4-300|300|300|300|300-uuuuu-5_3.json"
## [16] "alg1-prob-4-300|300|300|300|300-uuuuu-5_4.json"
## [17] "alg1-prob-4-300|300|300|300|300-uuuuu-5_5.json"
## [18] "alg1-prob-4-50|50|50|50|50-lllll-5_1.json"     
## [19] "alg1-prob-4-50|50|50|50|50-lllll-5_2.json"     
## [20] "alg1-prob-4-50|50|50|50|50-lllll-5_3.json"     
## [21] "alg1-prob-4-50|50|50|50|50-lllll-5_4.json"     
## [22] "alg1-prob-4-50|50|50|50|50-lllll-5_5.json"     
## [23] "alg1-prob-5-100|100-uu-2_2.json"               
## [24] "alg1-prob-5-100|100-uu-2_3.json"               
## [25] "alg1-prob-5-100|100|100|100|100-lllll-5_1.json"
## [26] "alg1-prob-5-100|100|100|100|100-lllll-5_2.json"
## [27] "alg1-prob-5-100|100|100|100|100-lllll-5_3.json"
## [28] "alg1-prob-5-100|100|100|100|100-lllll-5_4.json"
## [29] "alg1-prob-5-100|100|100|100|100-lllll-5_5.json"
## [30] "alg1-prob-5-100|100|100|100|100-mmmmm-5_5.json"
## [31] "alg1-prob-5-100|100|100|100|100-uuuuu-5_1.json"
## [32] "alg1-prob-5-100|100|100|100|100-uuuuu-5_4.json"
## [33] "alg1-prob-5-100|100|100|100|100-uuuuu-5_5.json"
## [34] "alg1-prob-5-200|200|200|200-llll-4_1.json"     
## [35] "alg1-prob-5-200|200|200|200-llll-4_2.json"     
## [36] "alg1-prob-5-200|200|200|200-llll-4_3.json"     
## [37] "alg1-prob-5-200|200|200|200-llll-4_4.json"     
## [38] "alg1-prob-5-200|200|200|200-llll-4_5.json"     
## [39] "alg1-prob-5-200|200|200|200-mmmm-4_1.json"     
## [40] "alg1-prob-5-200|200|200|200-mmmm-4_2.json"     
## [41] "alg1-prob-5-200|200|200|200-mmmm-4_3.json"     
## [42] "alg1-prob-5-200|200|200|200-mmmm-4_4.json"     
## [43] "alg1-prob-5-200|200|200|200-mmmm-4_5.json"     
## [44] "alg1-prob-5-200|200|200|200-uull-4_1.json"     
## [45] "alg1-prob-5-200|200|200|200-uull-4_2.json"     
## [46] "alg1-prob-5-200|200|200|200-uull-4_3.json"     
## [47] "alg1-prob-5-200|200|200|200-uull-4_4.json"     
## [48] "alg1-prob-5-200|200|200|200-uull-4_5.json"     
## [49] "alg1-prob-5-200|200|200|200|200-lllll-5_1.json"
## [50] "alg1-prob-5-200|200|200|200|200-lllll-5_2.json"
## [51] "alg1-prob-5-200|200|200|200|200-lllll-5_3.json"
## [52] "alg1-prob-5-200|200|200|200|200-lllll-5_4.json"
## [53] "alg1-prob-5-200|200|200|200|200-lllll-5_5.json"
## [54] "alg1-prob-5-300|300|300|300-llll-4_1.json"     
## [55] "alg1-prob-5-300|300|300|300-llll-4_2.json"     
## [56] "alg1-prob-5-300|300|300|300-llll-4_3.json"     
## [57] "alg1-prob-5-300|300|300|300-llll-4_4.json"     
## [58] "alg1-prob-5-300|300|300|300-llll-4_5.json"     
## [59] "alg1-prob-5-300|300|300|300-mmmm-4_1.json"     
## [60] "alg1-prob-5-300|300|300|300-mmmm-4_2.json"     
## [61] "alg1-prob-5-300|300|300|300-mmmm-4_3.json"     
## [62] "alg1-prob-5-300|300|300|300-mmmm-4_4.json"     
## [63] "alg1-prob-5-300|300|300|300-mmmm-4_5.json"     
## [64] "alg1-prob-5-300|300|300|300-uull-4_1.json"     
## [65] "alg1-prob-5-300|300|300|300-uull-4_2.json"     
## [66] "alg1-prob-5-300|300|300|300-uull-4_3.json"     
## [67] "alg1-prob-5-300|300|300|300-uull-4_4.json"     
## [68] "alg1-prob-5-300|300|300|300-uull-4_5.json"     
## [69] "alg1-prob-5-50|50|50|50|50-lllll-5_1.json"     
## [70] "alg1-prob-5-50|50|50|50|50-lllll-5_2.json"     
## [71] "alg1-prob-5-50|50|50|50|50-lllll-5_3.json"     
## [72] "alg1-prob-5-50|50|50|50|50-lllll-5_4.json"     
## [73] "alg1-prob-5-50|50|50|50|50-lllll-5_5.json"

463/1162 classified files have not been fully classified (only classified extreme).

Problems solved for the analysis

Note that the width of objective \(i = 1, \ldots p\), \(w_i = [l_i, u_i]\) should be approx. \(10000m\). Check:

## # A tibble: 4 × 6
##       m mean_width1 mean_width2 mean_width3 mean_width4 mean_width5
##   <dbl>       <dbl>       <dbl>       <dbl>       <dbl>       <dbl>
## 1     2      19245.      19221.      19213.      18996.      18690.
## 2     3      28760.      28800.      28689.      28479.      27847.
## 3     4      38302.      38348.      38158.      37758.      36803.
## 4     5      47840.      47994.      47848.      47431       44509.

Size of \(Y_N\)

What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?

## # A tibble: 4 × 3
##   method mean_card     n
##   <chr>      <dbl> <int>
## 1 l       1506031.   315
## 2 m        572395.   310
## 3 u        109824.   310
## 4 ul       206763.   300

Does \(p\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     p mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     8293.    80
##  2 m          2     6828.    80
##  3 u          2     1164.    80
##  4 ul         2     2920.    80
##  5 l          3   148913.    80
##  6 m          3   180435.    80
##  7 u          3    12475.    80
##  8 ul         3    26863.    80
##  9 l          4  1286899.    80
## 10 m          4  1063823.    80
## 11 u          4   110045.    80
## 12 ul         4   267769.    80
## 13 l          5  4784950.    75
## 14 m          5  1105081.    70
## 15 u          5   345012.    70
## 16 ul         5   637081.    60

Does \(m\) have an effect?

## # A tibble: 16 × 4
## # Groups:   method [4]
##    method     m mean_card     n
##    <chr>  <dbl>     <dbl> <int>
##  1 l          2     8173.    80
##  2 m          2     5688.    80
##  3 u          2     4201.    80
##  4 ul         2     4923.    80
##  5 l          3   166384.    80
##  6 m          3    90077.    80
##  7 u          3    37283.    80
##  8 ul         3    90425.    80
##  9 l          4  1596091.    80
## 10 m          4   874692.    80
## 11 u          4   190675.    80
## 12 ul         4   485509.    80
## 13 l          5  4436637.    75
## 14 m          5  1425800.    70
## 15 u          5   221040.    70
## 16 ul         5   259341.    60

Let us try to fit the results using function \(y=c_1 s^{(c_2p)} m^{c_3p}\) (different functions was tried and this gave the highest \(R^2\)) for each method.

## # A tibble: 4 × 15
##   method fit    tidied   r.squared adj.r.squared sigma statistic   p.value    df
##   <chr>  <list> <list>       <dbl>         <dbl> <dbl>     <dbl>     <dbl> <dbl>
## 1 l      <lm>   <tibble>     0.856         0.855 1.05       929. 3.90e-132     2
## 2 m      <lm>   <tibble>     0.768         0.766 1.25       507. 5.07e- 98     2
## 3 ul     <lm>   <tibble>     0.903         0.903 0.747     1389. 1.89e-151     2
## 4 u      <lm>   <tibble>     0.947         0.947 0.527     2759. 6.52e-197     2
## # ℹ 6 more variables: logLik <dbl>, AIC <dbl>, BIC <dbl>, deviance <dbl>,
## #   df.residual <int>, nobs <int>

## # A tibble: 4 × 4
##   method    c1     c2    c3
##   <chr>  <dbl>  <dbl> <dbl>
## 1 l       83.0 0.0836 1.24 
## 2 m       89.2 0.0847 1.11 
## 3 ul      30.1 0.117  1.12 
## 4 u       23.5 0.135  0.955

Relative size of \(Y_N\)

Nondominated points classification

We classify the nondominated points into, extreme, supported non-extreme and unsupported.

## # A tibble: 1 × 3
##   minPctEx avePctExt maxPctEx
##      <dbl>     <dbl>    <dbl>
## 1 0.000461    0.0449    0.330

## # A tibble: 4 × 4
##   method minPctEx avePctExt maxPctEx
##   <chr>     <dbl>     <dbl>    <dbl>
## 1 l      0.00443     0.0761    0.302
## 2 ul     0.00635     0.0719    0.330
## 3 m      0.000461    0.0205    0.147
## 4 u      0.00196     0.0132    0.104

Plots used in the paper